Minimum-Action Learning: Energy-Constrained Symbolic Model Selection for Physical Law Identification from Noisy Data
This work addresses the challenge of interpretable model selection in scientific machine learning for physics applications, offering an incremental improvement over existing methods by integrating energy constraints and noise reduction.
The paper tackles the problem of identifying physical laws from noisy observational data by introducing Minimum-Action Learning (MAL), a framework that selects symbolic force laws from a basis library while enforcing energy conservation, achieving a 40% reduction in energy usage and 100% pipeline-level identification accuracy on benchmarks like Kepler gravity and Hooke's law.
Identifying physical laws from noisy observational data is a central challenge in scientific machine learning. We present Minimum-Action Learning (MAL), a framework that selects symbolic force laws from a pre-specified basis library by minimizing a Triple-Action functional combining trajectory reconstruction, architectural sparsity, and energy-conservation enforcement. A wide-stencil acceleration-matching technique reduces noise variance by 10,000x, transforming an intractable problem (SNR ~0.02) into a learnable one (SNR ~1.6); this preprocessing is the critical enabler shared by all methods tested, including SINDy variants. On two benchmarks -- Kepler gravity and Hooke's law -- MAL recovers the correct force law with Kepler exponent p = 3.01 +/- 0.01 at ~0.07 kWh (40% reduction vs. prediction-error-only baselines). The raw correct-basis rate is 40% for Kepler and 90% for Hooke; an energy-conservation-based criterion discriminates the true force law in all cases, yielding 100% pipeline-level identification. Basis library sensitivity experiments show that near-confounders degrade selection (20% with added r^{-2.5} and r^{-1.5}), while distant additions are harmless, and the conservation diagnostic remains informative even when the correct basis is absent. Direct comparison with noise-robust SINDy variants, Hamiltonian Neural Networks, and Lagrangian Neural Networks confirms MAL's distinct niche: interpretable, energy-constrained model selection that combines symbolic basis identification with dynamical rollout validation.